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Physics & Astronomy International Journal

Research Article Volume 2 Issue 3

Compact four–dimensional euclidean space embedded in the universe

Yoshimasa Kurihara

The High Energy Accelerator Organization (KEK), Japan

Correspondence: Yoshimasa Kurihara, The High Energy Accelerator Organization (KEK), Oho 1?1, Tsukuba, Ibaraki 305?0801, Japan, Tel 8129 8796-088

Received: June 06, 2018 | Published: June 19, 2018

Citation: Kurihara Y. Compact four-dimensional euclidean space embedded in the universe. Phys Astron Int J. 2018;2(3):237-241. DOI: 10.15406/paij.2018.02.00092

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Abstract

A compact 4–dimensional manifold whose metric tensor has a positive determinant (named “Euclid ball”) is considered. The Euclid ball can be embedded in the Lorentzian space, and it exists stably through the history of the universe. Since the Euclid ball has the same metric as that of the Schwarzschild black hole on its 3–dimensional surface, asymptotic observers cannot distinguish them. Euclid balls can be created in the universe before the cosmological inflation (pre–universe). If large fraction of whole energy of the pre–universe was encapsulated in Euclid balls, they behave as the dark matter in the current universe. If Euclid balls formed at the end of the inflation with a heavy mass, they can be a seed of supper–massive black–holes, which are necessary to initiate a forming of galaxies in the early universe. The primordial gravitational wave and/or γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@ –ray bursts at early universe are a possible signal of the Euclidean ball.

Keywords: compact 4–dimensional manifold, euclid ball, lorentzian space, schwarzschild black hole, supper–massive black–holes

Introduction

The dark matter (DM) is, at first, introduced to explain an anomaly of stellar velocities in the galaxy.1 The existence of the DM is now widely accepted2 due to large number of observational supports.3,4 The DM is also necessary to explain galaxy formations in early universe.5 A 3–dimensional map of the DM distribution is made using a gravitational lensing effect.6 On the other hand, a direct observation of the DM is still not succeeded though number of experimental searches of the DM has been going on. Even though the cold DM density of the universe is precisely measured7,8 as Ω CDM =0.258(11) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHPo WvlmaaBaaajeaibaqcLbmacaqGdbGaaeiraiaab2eaaKqaGeqaaKqz GeGaaGypaiaaicdacaaIUaGaaGOmaiaaiwdacaaI4aGaaGikaiaaig dacaaIXaGaaGykaaaa@43FE@ , still it is not identified what it is experimentally. Several candidates of the cold DM are proposed, such as the lightest particle in the SUSY model (LSP), axions9,10 introduced to solve the strong CP problem, primordial black holes,11 and so on. Though recent direct searches of the SUSY particles at the LHC experiments report negative results, still a weak interacting LSP can be the DM candidate. Astroparticle discussions give additional constraint12,13 for the SUSY model to avoid the overproduction of the SUSY–DM. Each scenario has its advantages and disadvantages, and is not conclusive yet. In this report, we propose new candidate of the DM based on the geometrical model including quantum effects of the general relativity.

Euclid–ball solution

A 4–dimensional smooth manifold can be categorised by a sign of a determinant of their metric tensor. It is referred to as a metric–sign hereafter. If a manifold is diffeomorphism to the Euclid space, whose metric tensor is ζ =diag(1,1,1,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcqaH2oGEjuaGdaWgaaqcbasaaKqzadGaeyOiGCRaeyOiGClaleqa aKqzGeGaaGypaiaabsgacaqGPbGaaeyyaiaabEgacaaIOaGaaGymai aaiYcacaaIXaGaaGilaiaaigdacaaISaGaaGymaiaaiMcaaaa@49B5@ , it is called the Euclidean–space in this report, even if it is highly curved space. If the Eulidean–space is a compact, it is referred to as the Euclid–ball (EB). The smooth manifold, which is diffeomorphism to a space with a metric tensor η =diag(1,1,1,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAlmaaBaaajeaibaqcLbmacqGHIaYTcqGHIaYTaKqaGeqaaKqzGeGa aGypaiaabsgacaqGPbGaaeyyaiaabEgacaaIOaGaaGymaiaaiYcacq GHsislcaaIXaGaaGilaiabgkHiTiaaigdacaaISaGaeyOeI0IaaGym aiaaiMcaaaa@4B1A@ , is referred to as the Lorentzian space, again even if it is highly curved. Since a local space–time manifold without gravity is the (flat) Minkovski space and the general relativistic theory requires laws of physics must be diffeomophism, any space–time manifold in the universe is considered to be the Lorentzian space. It is follows from a fact that the metric–sign is the diffeomorphism invariance. While the Euclid ball cannot be created during the entire history of the universe after the inflation, there is a possibility that EBs were created before the inflation, due to a stochastic effect of the pre–universe.14 An example of a EB–solution is a space with a metric of;

d s E 2 = f E 2 (r)d t 2 + f E 2 (r)d r 2 + r 2 dΩ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaWGKbGaam4CaSWaa0baaKqaGeaajugWaiaabweaaKqaGeaajugW aiaaikdaaaqcLbsacaaI9aGaamOzaSWaa0baaKqaGeaajugWaiaabw eaaKqaGeaajugWaiaaikdaaaqcLbsacaaIOaGaamOCaiaaiMcacaWG KbGaamiDaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaS IaamOzaSWaa0baaKqaGeaajugWaiaabweaaKqaGeaajugWaiabgkHi TiaaikdaaaqcLbsacaaIOaGaamOCaiaaiMcacaWGKbGaamOCaSWaaW baaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaamOCaSWaaWba aKqaGeqabaqcLbmacaaIYaaaaKqzGeGaamizaiabfM6axbaa@61C9@
At r< r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGipaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaaaaa@3BCF@ , where f E 2 (r)= r c /r1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb WcdaqhaaqcbasaaKqzadGaaeyraaqcbasaaKqzadGaaGOmaaaajugi biaaiIcacaWGYbGaaGykaiaai2dacaWGYbWcdaWgaaqcbasaaKqzad Gaam4yaaqcbasabaqcLbsacaaIVaGaamOCaiabgkHiTiaaigdaaaa@46F7@  and dΩ=d θ 2 + sin 2 ϕ d ϕ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb GaeuyQdCLaaGypaiaadsgacqaH4oqClmaaCaaajeaibeqaaKqzadGa aGOmaaaajugibiabgUcaRKqbaoaavacakeqaleqajeaibaqcLbmaca aIYaaakeaajugibiGacohacaGGPbGaaiOBaaaacqaHvpGzcaaIGaGa amizaiabew9aMTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaaa@4DDE@ . The constant r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcKfaG=haajugWaiaadogaaKazba4=beaaaaa@3D98@ can be understood as the radius of the EB. One can confirm that scaler–and Ricci–curvature of this metric is zero. Therefore the EB metric is a solution of the Einstein equation in the Euclidean space with neither cosmological constant nor energy momentum tensor. Note that both of the Lorentzian and Euclidean spaces can be a solution of the Einstein equation with the null energy momentum tensor. The metric–sign is fixed by the energy momentum tensor and/or the boundary condition. For cosmological solutions, one usually requires the energy momentum tensor with Lorentzian metric and/or the asymptotic Minkovski space as the boundary condition. For instance, the Friedmann–Lmeître–Robertson–Walker (FLRW) metric1 with a positive curvature can be considered as a 3–dimensional sphere embedded in a 4–dimensional (flat) Euclid space. A region r<1/ K 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGipaiaaigdacaaIVaGaam4saSWaaWbaaKqaGeqabaqcLbmacaaI XaGaaG4laiaaikdaaaaaaa@3E3B@ is the Lorentzian–space, where K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb aaaa@3755@ is a curvature of the space.

The EB solution can be embedded in the Lorentzian–space by a following simple surgery: Preparing the Lorentzian–space with a Schwarzschild black hole whose metric as;

d s M 2 = f M 2 (r)d t 2 f M 2 (r)d r 2 r 2 dΩ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4CaSWaa0baaKqaGeaajugWaiaab2eaaKqaGeaajugWaiaaikda aaqcLbsacaaI9aGaamOzaSWaa0baaKqaGeaajugWaiaab2eaaKqaGe aajugWaiaaikdaaaqcLbsacaaIOaGaamOCaiaaiMcacaWGKbGaamiD aKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaeyOeI0Iaam OzaSWaa0baaKqaGeaajugWaiaab2eaaKqaGeaajugWaiabgkHiTiaa ikdaaaqcLbsacaaIOaGaamOCaiaaiMcacaWGKbGaamOCaSWaaWbaaK qaGeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0IaamOCaSWaaWbaaKqa GeqabaqcLbmacaaIYaaaaKqzGeGaamizaiabfM6axjaaiYcaaaa@624E@

where f M 2 =1 r c /r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb WcdaqhaaqcbasaaKqzadGaaeytaaqcbasaaKqzadGaaGOmaaaajugi biaai2dacaaIXaGaeyOeI0IaamOCaSWaaSbaaKqaGeaajugWaiaado gaaKqaGeqaaKqzGeGaaG4laiaadkhaaaa@44A3@ . The solution in a region r< r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGipaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaaaaa@3BCF@ is replaced by the EB, and adjust the angular coordinate θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp Gzaaa@384D@ at the surface r= r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaadkhajuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaaaaa@3C34@ .Two solutions give an identical metric at r= r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaaaaa@3BD0@  as d s E 2 (r= r c )=d s M 2 (r= r c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4CaSWaa0baaKqaGeaajugWaiaabweaaKqaGeaajugWaiaaikda aaqcLbsacaaIOaGaamOCaiaai2dacaWGYbqcfa4aaSbaaKqaGeaaju gWaiaadogaaSqabaqcLbsacaaIPaGaaGypaiaadsgacaWGZbWcdaqh aaqcbasaaKqzadGaaeytaaqcbasaaKqzadGaaGOmaaaajugibiaaiI cacaWGYbGaaGypaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeai beaajugibiaaiMcaaaa@53D8@ . The metric tensor is continuous (but not smooth) at the surface.

There is no singularity at the event horizon of the Schwarzschild black hole as know well. This fact can be seen if we look at the metric using (modified) Kruskal–Szekeres coordinates. In the region r> r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGOpaiaadkhajuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaaaaa@3C35@ , a coordinate is taken as;

U M = ( r r c 1 ) 1/2 e r/2 r c sinh( t 2 r c ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb WcdaWgaaqcbasaaKqzadGaaeytaaqcbasabaqcLbsacaaI9aqcfa4a aeWaaOqaaKqbaoaalaaakeaajugibiaadkhaaOqaaKqzGeGaamOCaS WaaSbaaKqaGeaajugWaiaadogaaKqaGeqaaaaajugibiabgkHiTiaa igdaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaaIXaGaaG 4laiaaikdaaaqcLbsacaWGLbWcdaahaaqcbasabeaajugWaiaadkha caaIVaGaaGOmaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibe aaaaqcLbsaciGGZbGaaiyAaiaac6gacaGGObqcfa4aaeWaaOqaaKqb aoaalaaakeaajugibiaadshaaOqaaKqzGeGaaGOmaiaadkhajuaGda WgaaqcbasaaKqzadGaam4yaaWcbeaaaaaakiaawIcacaGLPaaajugi biaaiYcaaaa@61CE@    (1)

V M = ( r r c 1 ) 1/2 e r/2 r c cosh( t 2 r c ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaSbaaKqaGeaajugWaiaab2eaaSqabaqcLbsacaaI9aqcfa4a aeWaaOqaaKqbaoaalaaakeaajugibiaadkhaaOqaaKqzGeGaamOCaK qbaoaaBaaajeaibaqcLbmacaWGJbaaleqaaaaajugibiabgkHiTiaa igdaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaaIXaGaaG 4laiaaikdaaaqcLbsacaWGLbWcdaahaaqcbasabeaajugWaiaadkha caaIVaGaaGOmaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibe aaaaqcLbsaciGGJbGaai4BaiaacohacaGGObqcfa4aaeWaaOqaaKqb aoaalaaakeaajugibiaadshaaOqaaKqzGeGaaGOmaiaadkhajuaGda WgaaqcbasaaKqzadGaam4yaaWcbeaaaaaakiaawIcacaGLPaaajugi biaaiYcaaaa@6292@ (2)

which gives a line element as;

d s M 2 =4 r c 3 r e r/ r c ( d U M 2 d V M 2 ) r 2 dΩ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4CaSWaa0baaKqaGeaajugWaiaab2eaaKqaGeaajugWaiaaikda aaqcLbsacaaI9aGaaGinaKqbaoaalaaakeaajugibiaadkhalmaaDa aajeaibaqcLbmacaWGJbaajeaibaqcLbmacaaIZaaaaaGcbaqcLbsa caWGYbaaaiaadwgalmaaCaaajeaibeqaaKqzadGaeyOeI0IaamOCai aai+cacaWGYbWcdaWgaaqcbasaaKqzadGaam4yaaqcbasabaaaaKqb aoaabmaakeaajugibiaadsgacaWGvbWcdaqhaaqcbasaaKqzadGaae ytaaqcbasaaKqzadGaaGOmaaaajugibiabgkHiTiaadsgacaWGwbWc daqhaaqcbasaaKqzadGaaeytaaqcbasaaKqzadGaaGOmaaaaaOGaay jkaiaawMcaaKqzGeGaeyOeI0IaamOCaSWaaWbaaKqaGeqabaqcLbma caaIYaaaaKqzGeGaamizaiabfM6axjaai6caaaa@68BD@             (3)

In this coordinate, r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ can be written as a function of U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb aaaa@375F@ and V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb aaaa@3760@ . In the region r> r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGOpaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaaaaa@3BD1@ , the Kruskal–Szekeres coordinates are modified as;

U E = ( 1 r r c ) 1/2 e r/2 r c sin( tan 1 (t/ r c ) 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb WcdaWgaaqcbasaaKqzadGaaeyraaqcbasabaqcLbsacaaI9aqcfa4a aeWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibiaadk haaOqaaKqzGeGaamOCaKqbaoaaBaaaleaajugibiaadogaaSqabaaa aaGccaGLOaGaayzkaaWcdaahaaqcbasabeaajugWaiaaigdacaaIVa GaaGOmaaaajugibiaadwgalmaaCaaajeaibeqaaKqzadGaamOCaiaa i+cacaaIYaGaamOCaSWaaSbaaKqaGeaajugWaiaadogaaKqaGeqaaa aajugibiGacohacaGGPbGaaiOBaKqbaoaabmaakeaajuaGdaWcaaGc baqcfa4aaubiaOqabSqabKqaGeaajugWaiabgkHiTiaaigdaaOqaaK qzGeGaciiDaiaacggacaGGUbaaaiaaiIcacaWG0bGaaG4laiaadkha lmaaBaaajeaibaqcLbmacaWGJbaajeaibeaajugibiaaiMcaaOqaaK qzGeGaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaaGilaaaa@6985@         (4)

V E = ( 1 r r c ) 1/2 e r/2 r c cos( tan 1 (t/ r c ) 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb WcdaWgaaqcbasaaKqzadGaaeyraaqcbasabaqcLbsacaaI9aqcfa4a aeWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibiaadk haaOqaaKqzGeGaamOCaKqbaoaaBaaajeaibaqcLbmacaWGJbaaleqa aaaaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaaIXaGaaG 4laiaaikdaaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiaa dkhacaaIVaGaaGOmaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaaje aibeaaaaqcLbsaciGGJbGaai4BaiaacohajuaGdaqadaGcbaqcfa4a aSaaaOqaaKqbaoaavacakeqaleqajeaibaqcLbmacqGHsislcaaIXa aakeaajugibiGacshacaGGHbGaaiOBaaaacaaIOaGaamiDaiaai+ca caWGYbWcdaWgaaqcbasaaKqzadGaam4yaaqcbasabaqcLbsacaaIPa aakeaajugibiaaikdaaaaakiaawIcacaGLPaaajugibiaaiYcaaaa@6ACD@          (5)

which gives a line element as;

d s E 2 =4 r c 3 r ( 1+ t 2 r c 2 ) e r/ r c ( d U E 2 +d V E 2 )+ r 2 dΩ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaWGKbGaam4CaSWaa0baaKqaGeaajugWaiaabweaaKqaGeaajugW aiaaikdaaaqcLbsacaaI9aGaaGinaKqbaoaalaaakeaajugibiaadk halmaaDaaajeaibaqcLbmacaWGJbaajeaibaqcLbmacaaIZaaaaaGc baqcLbsacaWGYbaaaKqbaoaabmaakeaajugibiaaigdacqGHRaWkju aGdaWcaaGcbaqcLbsacaWG0bWcdaahaaqcbasabeaajugWaiaaikda aaaakeaajugibiaadkhalmaaDaaajeaibaqcLbmacaWGJbaajeaiba qcLbmacaaIYaaaaaaaaOGaayjkaiaawMcaaKqzGeGaamyzaKqbaoaa CaaaleqajeaibaqcLbmacqGHsislcaWGYbGaaG4laiaadkhalmaaBa aajeaibaqcLbmacaWGJbaajeaibeaaaaqcfa4aaeWaaOqaaKqzGeGa amizaiaadwfalmaaDaaajeaibaqcLbmacaqGfbaajeaibaqcLbmaca aIYaaaaKqzGeGaey4kaSIaamizaiaadAfalmaaDaaajeaibaqcLbma caqGfbaajeaibaqcLbmacaaIYaaaaaGccaGLOaGaayzkaaqcLbsacq GHRaWkcaWGYbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsa caWGKbGaeuyQdCLaaGOlaaaa@7A00@            (6)

At r= r c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaaaaa@3BD0@ , both solutions coincide as ( U M , V M )=( U E , V E )=(0,0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIOa GaamyvaKqbaoaaBaaajeaibaqcLbmacaqGnbaaleqaaKqzGeGaaGil aiaadAfajuaGdaWgaaqcbasaaKqzadGaaeytaaWcbeaajugibiaaiM cacaaI9aGaaGikaiaadwfalmaaBaaajeaibaqcLbmacaqGfbaajeai beaajugibiaaiYcacaWGwbqcfa4aaSbaaKqaGeaajugWaiaabweaaS qabaqcLbsacaaIPaGaaGypaiaaiIcacaaIWaGaaGilaiaaicdacaaI Paaaaa@5092@ . New coordinates are also continuous in entire region of the space as shown in Figure 1. The Euclidean Kruskal–Szekeres coordinates are similar to the Lorentzian ones at the inside of the EB. The Penrose diagram is shown in Figure 2. A region below a 45 degree line is a Lorentzian solution of a Schwarzschild black hole. A standard transformation from Kruskal–Szekeres coordinates to Penrose one (U,V) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIOa GaamyvaiaaiYcacaWGwbGaaGykaaaa@3A55@  is used in this region. In the Euclidean region, we set ( U E , V E )=(U,V) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIOa GaamyvaKqbaoaaBaaajeaibaqcLbmacaqGfbaaleqaaKqzGeGaaGil aiaadAfajuaGdaWgaaqcbasaaKqzadGaaeyraaWcbeaajugibiaaiM cacaaI9aGaaGikaiaadwfacaaISaGaamOvaiaaiMcaaaa@45BE@ . A centre of the EB, r=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaaicdaaaa@38FD@ , is a circle with unit radius, and thus there are no world lines connected from inside of the EB to the asymptotic observer at r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaeyOKH4QaeyOhIukaaa@3ADA@ . Therefore, the EB cannot be distinguished from a Schwarzschild black hole with a mass M= r c /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb GaaGypaiaadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaajugi biaai+cacaaIYaaaaa@3DAF@ .

Figure 1 Two components of (modified) Kruskal–Szekeres coordinates, U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb aaaa@375F@ >and V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb aaaa@3760@ are shown with r c =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWajjugibi aadkhalmaaBaaajeaibaqcLbmacaWGJbaajeaibeaajugibiaai2da caaIXaaaaa@3C56@ at t=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b GaaGypaiaaigdaaaa@3900@ . Solid (dotted) lines show the U M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb WcdaWgaaqcbasaaKqzadGaaeytaaqcbasabaaaaa@39DD@ – ( V M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaSbaaKqaGeaajugWaiaab2eaaSqabaaaaa@3A42@ –) coordinate, respectively. In the Euclidean–region r< r c =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGipaiaadkhajuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaajugi biaai2dacaaIXaaaaa@3E44@ , U E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb WcdaWgaaqcbasaaKqzadGaaeyraaqcbasabaaaaa@39D5@  and V E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb WcdaWgaaqcbasaaKqzadGaaeyraaqcbasabaaaaa@39D6@ are shown by dashed and dot–dashed lines, respectively.

Figure 2 The Penrose diagram for the embedded solution. Solid (dotted) lines show the r= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaaaa@3843@  ( t= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b GaaGypaaaa@3845@ ) constant line, respectively. Only a region t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b GaaGOpaiaaicdaaaa@3900@ including an asymptotic observer is drawn in the figure. The 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaI0a GaaGynaaaa@3802@ degree line shows the event horizon, which is r=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaaigdaaaa@38FE@  and t=+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b GaaGypaiabgUcaRiabg6HiLcaa@3A98@ .


1see, for instance,15

Euclidian–balls and cosmological inflation

A creation of EBs in the Lorentzian universe may be possible by taking account of a stochastic effect of the general relativity.14 Under the stochastic quantization, classical solutions of the general relativity can be obtained by taking an ensemble average of the space time manifold. Ensembles of the space–time manifold can be expressed as = + ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifjaai2da cqWFWesrlmaaBaaajeaibaqcLbmacqGHRaWkaKqaGeqaaKqzGeGaey yLIuSae8hmHu0cdaWgaaqcbasaaKqzadGaeyOeI0cajeaibeaajugi biabgwPiflab=btifLqbaoaaBaaajeaibaqcLbmacqGHXcqSaSqaba aaaa@539B@ , where + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifLqbaoaa BaaajeaibaqcLbmacqGHRaWkaSqabaaaaa@441E@ ( MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifTWaaSba aKqaGeaajugWaiabgkHiTaqcbasabaaaaa@43C5@ ) is a set of the Euclidean–(Lorentzian–)space, respectively. A third ensemble ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifTWaaSba aKqaGeaajugWaiabgglaXcqcbasabaaaaa@44C6@ is a set of Lorentzian–manifolds in which the Euclidean–manifold is embedded, and vice versa. An element of ensembles U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=rr8vLqbaoaa BaaajeaibaqcLbmacqGHIaYTaSqabaqcLbsacqGHiiIZcqWFWesrju aGdaWgaaqcbasaaKqzadGaeyOiGClaleqaaaaa@4C4A@ is referred to as the “sample universe”, and ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifTWaaSba aKqaGeaajugWaiabgglaXcqcbasabaaaaa@44C6@ ( U ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaajugibiab=rr8vTWaaSba aKqaGeaajugWaiabgglaXcqcbasabaaaaa@458A@ ) are called “chimera ensemble (chimera space)”, respectively. An EB–solution in a FLRW universe is an example of the chimera universe. Each element of an ensemble is assumed to be labelled by a order parameter τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDcqGHiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb aiab=1risbaa@4486@ . A distance between two samples U (τ) U (τ)0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb sr4rNCHbacfaqcLbsacqWFLicutuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGGbaiab+rr8vLqbaoaaBaaajeaibaqcLbmacqGHIa YTaSqabaqcLbsacaaIOaGaeqiXdqNaaGykaiabgkHiTiab+rr8vLqb aoaaBaaajeaibaqcLbmacqGHIaYTaSqabaqcLbsacaaIOaGaeqiXdq NaaGykaiab=vIiqjabgwMiZkaaicdaaaa@5CAD@  is also introduced such as lim δ0 U (τ) U (τ+δ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaubeaO qabKqaGeaajugWaiabes7aKjabgkziUkaaicdaaSqabOqaaKqzGeGa ciiBaiaacMgacaGGTbaaaebbfv3ySLgzGueE0jxyaGqbaiab=vIiqn rr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacgaGae4hfXxvc fa4aaSbaaKqaGeaajugWaiabgkci3cWcbeaajugibiaaiIcacqaHep aDcaaIPaGaeyOeI0Iae4hfXxvcfa4aaSbaaKqaGeaajugWaiabgkci 3cWcbeaajugibiaaiIcacqaHepaDcqGHRaWkcqaH0oazcaaIPaGae8 xjIaLaaGypaiaaicdaaaa@6784@ . The order parameter is not a component of the space time coordinate (time in a standard sense), but an independent parameter from the coordinate, such as the fictitious time in the stochastic quantization by Parisi & Wu16.

Suppose the space time manifold of the universe was entirely Euclidean before the inflation as discussed by Hawking & Hartle17 and had only one scalar filed in it. This universe is referred to as a pre–universe hereafter. The pre–universe is assumed to be a compact and small. The scalar field played a role to make the pre–universe a compact, and to fix the scale of the pre–universe. (If there is only the space time without any fields, one cannot measure a distance.) The energy momentum tensor of the perfect fluid in a Lorentzian–space is T M =diag( c 2 ρ,P,P,P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaab2eaaSqabaqcLbsacaaI9aGaaeiz aiaabMgacaqGHbGaae4zaiaaiIcacaWGJbqcfa4aaWbaaSqabKqaGe aajugWaiaaikdaaaqcLbsacqaHbpGCcaaISaGaeyOeI0Iaamiuaiaa iYcacqGHsislcaWGqbGaaGilaiabgkHiTiaadcfacaaIPaaaaa@4E0A@ , and it in the Euclidean–space is T E =diag( c 2 ρ,P,P,P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqcbasaaKqzadGaaeyraaqcbasabaqcLbsacaaI9aGaaeiz aiaabMgacaqGHbGaae4zaiaaiIcacaWGJbqcfa4aaWbaaSqabKqaGe aajugWaiaaikdaaaqcLbsacqaHbpGCcaaISaGaamiuaiaaiYcacaWG qbGaaGilaiaadcfacaaIPaaaaa@4AD7@ , where ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCaaa@3845@ and P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb aaaa@375A@ are a density and pressure of fluid, respectively. Physical constants are written explicitly in this section. From the Friedmann equation for the Lorentzian space, the Hubble parameter H M (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqcbasaaKqzadGaaeytaaqcbasabaqcLbsacaaIOaGaamiD aiaaiMcaaaa@3CBD@ follows an equation of;

d H M dt := H ˙ M =4πG( 1+ P ρ c 2 )ρ+ K c 2 a 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIeajuaGdaWgaaqcbasaaKqzadGaaeytaaWc beaaaOqaaKqzGeGaamizaiaadshaaaGaaGOoaiaai2daceWGibGbai aalmaaBaaajeaibaqcLbmacaqGnbaajeaibeaajugibiaai2dacqGH sislcaaI0aGaeqiWdaNaam4raKqbaoaabmaakeaajugibiaaigdacq GHRaWkjuaGdaWcaaGcbaqcLbsacaWGqbaakeaajugibiabeg8aYjaa dogajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaaakiaawIcaca GLPaaajugibiabeg8aYjabgUcaRKqbaoaalaaakeGabaG6bKqzGeGa am4saiaadogajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaK qzGeGaamyyaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaaajugi biaaiYcaaaa@648C@        (7)

where G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb aaaa@3751@ is the Newtonian gravitational constant and a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb aaaa@376B@ is a scale function of t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b aaaa@377E@ . The curvature is assumed to be positive K>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb GaaGOpaiaaicdaaaa@38D7@ . Conversion of equation (7) from the Lorentzian space to Euclidean space can be done by a simple replacement of cic MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb GaeyOKH4QaamyAaiaadogaaaa@3B30@ as

d H E dt := H ˙ E =4πG( 1 P ρ c 2 )ρ K c 2 a 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIeajuaGdaWgaaqcbasaaKqzadGaaeyraaWc beaaaOqaaKqzGeGaamizaiaadshaaaGaaGOoaiaai2daceWGibGbai aalmaaBaaajeaibaqcLbmacaqGfbaajeaibeaajugibiaai2dacqGH sislcaaI0aGaeqiWdaNaam4raKqbaoaabmaakeaajugibiaaigdacq GHsisljuaGdaWcaaGcbaqcLbsacaWGqbaakeaajugibiabeg8aYjaa dogalmaaCaaajeaibeqaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPa aajugibiabeg8aYjabgkHiTKqbaoaalaaakeaajugibiaadUeacaWG JbWcdaahaaqcbasabeaajugWaiaaikdaaaaakeaajugibiaadggaju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaqcLbsacaaIUaaaaa@628A@          (8)

For the scalar field, P/(ρ c 2 )=1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb GaaG4laiaaiIcacqaHbpGCcaWGJbWcdaahaaqcbasabeaajugWaiaa ikdaaaqcLbsacaaIPaGaaGypaiaaigdacaaIVaGaaG4maaaa@41E8@ . Therefore, the scalar field in the Euclidean pre–universe induced positive pressure (from the internal observer point of view). During a period sampling the Euclidean–ensemble, the pre–universe stayed in the same scale. After enough duration of τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDaaa@384A@ , the pre–universe is expected to be uniform and isometric, and it must be highly curved due to the pressure of the scalar filed. If the Euclidean ensemble is connected to the chimera ensemble on some region of τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDaaa@384A@ , the pre–universe stared sampling universes in the chimera ensemble in some occasion. In a region of a Lorentzian space in the pre–universe, the scalar field caused an expanding force to the remnant, and then the pre–universe started an explosion inflationary.

On the other hand, the Euclidean region of the pre–universe stayed the same size. A quantitative sketch of a formation of the EBs is as follows (Figure 3): Suppose two Lorentzian space are embedded in the Euclidean space (Figure 3A). Each Lorentzian space had the FLRW metric with a positive curvature and had an event horizon at r=1/ K 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb GaaGypaiaaigdacaaIVaGaam4saSWaaWbaaKqaGeqabaqcLbmacaaI XaGaaG4laiaaikdaaaaaaa@3E3C@ . The metric is continuously connected to the Euclidean space at the surfaces. At first distance between two surfaces was much larger than the Planck length. When two surfaces of the Lorentzian space approached within an order of the Planck length, the quantum fluctuation made clacks in the Euclidean–space (Figure 3B) and two Lorentzian regions merged together. Euclidean spaces embedded in the Lorentzian space became 3–dimensional ball due to the scalar field encapsulated in them (Figure 3C). Therefore EBs is distributed on the boundary of two Lorentzian region (named Via). At the Euclidean region among three or more Lorentzian region (named Piazza), larger EBs can be formed as shown in Figure 3D. When the inflation period was terminated, EBs are distributed at the via (small EBs) and the Piazza (large EBs). Cosmic voids are corresponding to a central part of the Lorentzian space in the pre–universe just before the inflation. In this scenario, requirements for the –folding due to the inflation may be relaxed compared from the standard scenario, because the pre–universe was already in the thermal equilibrium before the inflation. However the inflation itself is necessary to realize the flat universe form the highly curved pre–universe. After the inflation, the energy of the scalar field in the Lorentzian region is converted to visible matter/gauge fields during the big bang. On the other hand, the energy stored in the EBs becomes invisible other than the gravitational interaction.

Figure 3 A schematic drawing of a formation of HBs. (A) Close–up view of the Euclidean–space (shadowed area) sandwiched by two (expanding) Lorentzian space. (B) When a distance between two Lorentzian spaces became an order of the Planck length, Lorentzian spaces are bridged due to the quantum fluctuation. (C) EBs are formed on the surface between two Lorentzian spaces (Via) are collide each other. (D) In a region among three or more Lorentzian space (Piazza), lager EBs may be formed compared with those in the Via.

Indirect evidences of euclid–balls

While the EB is very similar to the primordial black hole, there is critical difference such that the EB cannot emit the Hawking radiation. A reason why the EB does not evaporate is simple: Although a virtual pair of quanta is created at the surface by the quantum effect of the field, none of them can go into the EB because of its metric, and thus it cannot propagate to the asymptotic observer, due to the energy momentum conservation. Therefore, the EB is a stable object and can play a role of the dark matter to contribute a formation of galaxies, and survived until the current universe. A total amount of EBs is free from strong constraints18 from astronomical observations. If a fraction of the energy which belongs to the Lorentzian region was Ω b / Ω CDM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHPo WvjuaGdaWgaaqcbasaaKqzadGaaeOyaaWcbeaajugibiaai+cacqqH PoWvlmaaBaaajeaibaqcLbmacaqGdbGaaeiraiaab2eaaKqaGeqaaa aa@41EB@ ,7,8 0.0484/0.258=0.187 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb sr4rNCHbacfaqcLbsacqWF8iIocaaIWaGaaGOlaiaaicdacaaI0aGa aGioaiaaisdacaaIVaGaaGimaiaai6cacaaIYaGaaGynaiaaiIdaca aI9aGaaGimaiaai6cacaaIXaGaaGioaiaaiEdaaaa@4987@  in the pre–universe, it is conserved until current universe, where Ω b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHPo WvlmaaBaaajeaibaqcLbmacaqGIbaajeaibeaaaaa@3AA6@ ( Ω CDM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHPo WvlmaaBaaajeaibaqcLbmacaqGdbGaaeiraiaab2eaaKqaGeqaaaaa @3C1E@ ) is a baryon (cold dark matter) density, respectively. Simple consequence of this model is that the dark matter cannot be observed by the detector whose target is weakly interacting particles or decay products of the dark matter, because the EB interacts with visible part of the universe only through the gravitational interaction.

Another consequence is the DM can have a mass spectrum, which means that heavy EBs can be produced as mentioned before. Recent observation reported that the almost all galaxies have a supper massive black hole (SMBH), which have a mass about 10 6 M 10 8 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa GaaGimaSWaaWbaaKqaGeqabaqcLbmacaaI2aaaaKqzGeGaamytaKqb aoaaBaaajeaibaqcLbmacqWIzkszaSqabaqeeuuDJXwAKbsr4rNCHb acfaqcLbsacqWF8iIocaaIXaGaaGimaSWaaWbaaKqaGeqabaqcLbma caaI4aaaaKqzGeGaamytaKqbaoaaBaaajeaibaqcLbmacqWIzkszaS qabaaaaa@4E8B@ , at the centre of the galaxy.19,20 Several formation mechanisms of such a heavy black hole are proposed, e.g., the SMBH are rapidly grown from the massive star of Pop III type21–23 in the early universe, or directory formed from dens gases.24–28 In both cases, an existence of the seed black hole with a mass about 10 M 100 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa GaaGimaiaad2ealmaaBaaajeaibaqcLbmacqWIzkszaKqaGeqaaebb fv3ySLgzGueE0jxyaGqbaKqzGeGae8hpIOJaaGymaiaaicdacaaIWa GaamytaSWaaSbaaKqaGeaajugWaiablMPiLbqcbasabaaaaa@48D3@ is necessary to from the SMBH. Moreover, the seed black hole must not be rotating, because an angular momentum disturbs a rapid growing of the black hole. A recent observation29 suggests the SMBH is appeared only 900Myr (500Myr) after the big bang (the Pop II, III star formation), respectively. Those observations are contradicted to the standard scenario of the Λ CMD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo atlmaaBaaajeaibaqcLbmacaqGdbGaaeytaiaabseaaKqaGeqaaaaa @3C05@ model.30 In the EB scenario, EBs may have a mass spectrum from the Planck mass to a certain upper bound. Moreover, EBs is expected to be created without an angular momentum, due to an isometry of the Euclidean–space in the pre–universe. Therefore, the EBs with a large diameter can be a seed of the SMBH. In order to estimate a mass spectrum of the EB created by the inflation of the Lorentzian part of the pre–universe, one has to solve the Einstein equation with a scalar filed in the chimera space, which is beyond the scope of this report.

Even if heavy EBs are formed as much as the stellar black holes, a observation of the EB is more difficult than that of the stellar black hole, because 1) EBs may not have a visible partner who supply hot gasses, and 2) EBs have small angular momentum. A possible method to observe EBs is to use the gravitational wave caused by a fusion of two ESs. There are several gravitational–wave observatories, such as advanced VIRGO,31 advanced LIGO,32 and KAGRA33. A critical (minimum) mass of the stellar black hole is estimated to be 2 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb sr4rNCHbacfaqcLbsacqWF8iIocaaIYaGaamytaKqbaoaaBaaaleaa jugWaiablMPiLbWcbeaaaaa@4170@ . If gravitational wave observatories will observe gravitational wave from a fusion of two black holes less than the critical mass, it is a candidate of a signal from the EB or primordial black hole. An observation of gravitational wave from merging two neutron stars (GW170817) is reported.34 Gravitational wave due to a fusion of binary neutron star can be distinguished from those from black holes (EBs). Another possible signal of gravitational wave from EBs may be hidden in a B–mode polarization of the CMB. EBs existed when the inflation was terminated and stared growing to the SMBH. Fusion of EBs started immediately after the inflation and emitted gravitational wave. This gravitational wave overlaps the primordial gravitational wave2 produced during the inflation and put fingerprints on the CMB.

If the EB captured a source of hot gasses, a direct observation of the EB as the black hole with mass less than the critical mass is also possible. A small radius EBs (smaller than the critical mass) can emit a large total luminosity, because the hot gasses can go to a stronger gravity region than for stellar black holes before reaching a event horizon. The smallest black hole ever observed is XTE J1650500, whose mass is estimated to be 2.7 M 7.3 M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIYa GaaGOlaiaaiEdacaWGnbWcdaWgaaqcbasaaKqzadGaeSyMIugajeai beaarqqr1ngBPrgifHhDYfgaiuaajugibiab=XJi6iaaiEdacaaIUa GaaG4maiaad2ealmaaBaaajeaibaqcLbmacqWIzkszaKqaGeqaaaaa @499A@ .36
Another candidate to observe the EB is the γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@ –ray burst (GRB), especially those with large redshift. Possible aspects of a black hole burst are discussed.37 The Swift satellite confirmed eight GRBs with z>6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6b GaaGOpaiaaiAdaaaa@390C@ during its ten years operation3. A rate of the GRB at z>6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6b GaaGOpaiaaiAdaaaa@390C@ is estimated 0.6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb sr4rNCHbacfaqcLbsacqWF8iIocaaIWaGaaGOlaiaaiAdaaaa@3E70@ bursts yr–1sr–1.38 There is possibility that a part of those GRBs comes from a fusion of the EB and other stellar objects, such as nuclear star or massive gas stars.


2 See, for example, a summary report35 and references there in.

3 A list of GRBs with z>6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG6b GaaGOpaiaaiAdaaaa@390C@ and their detailed discussions can be found in Salvaterra R, et al.38.

Discussions and Summary

We introduced the EBs as a new candidate of the dark matter, that is four–dimensional Euclid–space embed in the universe. The EB solution is very similar to a Schwarzschild black hole, and also to the FLRW solution. The FLRW solution is the Lorentzian solution embedded in the Euclidean space, and the EB solution is vice versa. If the pre–universe before the cosmological inflation had the Euclidean metric,14,17 EBs can be created during the inflation and stably survived at the current universe. According to the model proposed in this study, the universe can be understood as the chimera space including large amount of EBs in the almost flat Lorentzian space. This model potentially explains an origin of the DM, the homogeneity problem of the universe, formation of the SMBH, and an origin of the GRBs with large redshift.

Acknowledgements

I appreciate the kind hospitality of all members of the theory group of Nikhef, particularly Prof. J.Vermaseren and Prof. E. Laenen. A major part of this study has been conducted during my stay at Nikhef in 2016 and 2017. In addition, I would like to thank Dr. Y. Sugiyama for his continuous encouragement and fruitful discussion.

Conflict of interest

Author declares there is no conflict of interest.

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